|
%I #21 Jul 14 2018 15:37:21
%S 0,3,26,442,12899,582381,37700452,3315996468,380835212037,
%T 55380159334315,9950025870043126,2165134468142294430,
%U 561245519520167902471,170913803045738754172185,60421582956702701927410120,24543570079301728283314502248,11353373604627607560431407875081
%N Number of proper mergings of an n-antichain and an n-chain.
%C a(n) is also the number of monotone (n+1)-colorings of a complete bipartite digraph K(n,n), where a monotone (n+1)-coloring is a labeling w of the vertices of K(n,n) with integers in {1,2,...,n+1} such that for every arc (e1, e2) we have w(e1) <= w(e2).
%H H. Mühle, <a href="http://dx.doi.org/10.1016/j.disc.2014.03.020">Counting Proper Mergings of Chains and Antichains</a>, Discrete Math., Vol. 327(C), 2014, 118-129. Also <a href="https://arxiv.org/abs/1206.3922">arXiv:1206.3922 [math.CO]</a>, 2012.
%F a(n) = Sum_{i=1..n+1} ((n+2-i)^n - (n+1-i)^n)*i^n.
%e For n=1, the three proper mergings of a 1-chain {x} and a 1-antichain {y} are x<y, y<x, and x,y.
%p a := n -> add(((n-i+1)^n-(n-i)^n)*(i+1)^n, i=0..n):
%p seq(a(n), n=0..16); # _Peter Luschny_, Nov 11 2016
%t a[0] = 0; a[n_] := Sum[((n-i+1)^n - (n-i)^n)*(i+1)^n, {i, 0, n}];
%t Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Jul 14 2018, after _Peter Luschny_ *)
%o (PARI) a(n) = sum(i=1, n+1, ((n+2-i)^n - (n+1-i)^n)*i^n); \\ _Michel Marcus_, Jul 14 2018
%Y Cf. A085465.
%K nonn
%O 0,2
%A _Henri Mühle_, Nov 11 2016
|
